Rolling Regressions

Imports

Summary

In this assignment we calculate regression coefficients for daily returns of 210 individual equities versus the market derived from exponentially weighted and boxcar widows of various lengths and compare them to coefficients calculated from a forward boxcar window. Our basis for comparison is that the coefficients calculated from the forward boxcar window are the "correct" coefficients and we therefore endeavor to determine which set of historical coefficients are most similar. To determine similarity we perform the following analyses across window methodology and length.

Returns are calculated from daily adjusted closing prices for the period from 2016-01-01 to 2020-12-31 for 210 equities that meet the criteria from the quantile trading assignment. We establish weights for the exponentially weighted windows based on characteristic times of $\frac{1}{\lambda}$ ranging from 3 to 60 observations (our analysis makes the simplifying assumption that the duration between observations is the same). Historical boxcar windows lengths are calculated as $\frac{2}{\lambda}$ and range from 6 to 120 observations. In order to have one historical boxcar window length with the same length as the forward window, we set the length of the forward boxcar window at 6 observations instead of 5.

Analysis

Load Prices

Compare Weights

The chart below shows the normalized cumulative weight for each of the exponentially weighted window lengths through 120 days. The main point is that observations well past the time $t$ continue to impact the moving average. In general, observations greater than $t$ comprise approximately 35% of the statistics. Given a boxcar window length of $\frac{2}{\lambda}$ for 10 observations of 20, for example ($\lambda = 0.1)$, the equivalent exponentially weighted window affords approximately 13% of its weight to observations greater than 20.

Calculate Returns

Here is our familiar pattern of generally upward drift with a significant drop and increased volatility at the onset of the pandemic. It's interesting that the performance of this particular set of equities was so much worse than the market over this period.

Calculate Betas

Here we calculate betas for each equity for each observation for each window type for each window length (1.3 million in total). We do this by using the rolling covariance matrix function in pandas to get the covariance versus SPY and the variance for each ticker and then calculate betas directly as $\hat{\beta} = \frac{\mathrm{Cov}(x,y)}{\mathrm{Var}(x)}$.

We do a quick check of the the mean $\hat{\beta}$ for the forward boxcar window to make sure data is available for each of our selected tickers.

Compare Moments

This is a dense chart, but allows us to compare the betas for the various combinations of window type and window length from one chart with y_axes for each statistic on the same scale. The moments are across the columns and the rows by window type. Note that the window length for all of the length labels for the forward box car window is 6 observations to facilitate comparison with the other window methods at varying lengths.

Compare Distributions

Here we plot distributions of coefficients for each window length. The range has been limited to betas of with an absolute value less than two in order to make differences in order to focus on differences in the core distributions, although there are outliers with absolute values greater than 8 in the t_03 window duration. Note again, that the forward boxcar window length is the same in each plot at 6 observations.

Compare Correlations

Lastly, we analyze correlation between the series of coefficients. If we were concerned with matching the series of coefficients, i.e., the ewm and boxcar series for a particular window length to the next forward boxcar window, the merely analyzing the distributions would be insufficient since the distributions do not take order into account. As a case in point, the distributions of the forward boxcar window and rearward looking boxcar window of 6 observations are nearly identical, yet when we look at the correlation below, we see that they are the least correlated. The highest correlation was actually with the exponentially weighted window of 30 days with a correlation of 0.326.